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Tag 0791

Chapter 7: Sites and Sheaves > Section 7.29: Localization of topoi

Lemma 7.29.3. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $\mathcal{C}/\mathcal{F}$ be the category of pairs $(U, s)$ where $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$ and $s \in \mathcal{F}(U)$. Let a covering in $\mathcal{C}/\mathcal{F}$ be a family $\{(U_i, s_i) \to (U, s)\}$ such that $\{U_i \to U\}$ is a covering of $\mathcal{C}$. Then $j : \mathcal{C}/\mathcal{F} \to \mathcal{C}$ is a continuous and cocontinuous functor of sites which induces a morphism of topoi $j : \mathop{\textit{Sh}}\nolimits(\mathcal{C}/\mathcal{F}) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C})$. In fact, there is an equivalence $\mathop{\textit{Sh}}\nolimits(\mathcal{C}/\mathcal{F}) = \mathop{\textit{Sh}}\nolimits(\mathcal{C})/\mathcal{F}$ which turns $j$ into $j_\mathcal{F}$.

Proof. We omit the verification that $\mathcal{C}/\mathcal{F}$ is a site and that $j$ is continuous and cocontinuous. By Lemma 7.20.5 there exists a morphism of topoi $j$ as indicated, with $j^{-1}\mathcal{G}(U, s) = \mathcal{G}(U)$, and there is a left adjoint $j_!$ to $j^{-1}$. A morphism $\varphi : * \to j^{-1}\mathcal{G}$ on $\mathcal{C}/\mathcal{F}$ is the same thing as a rule which assigns to every pair $(U, s)$ a section $\varphi(s) \in \mathcal{G}(U)$ compatible with restriction maps. Hence this is the same thing as a morphism $\varphi : \mathcal{F} \to \mathcal{G}$ over $\mathcal{C}$. We conclude that $j_!* = \mathcal{F}$. In particular, for every $\mathcal{H} \in \mathop{\textit{Sh}}\nolimits(\mathcal{C}/\mathcal{F})$ there is a canonical map $$ j_!\mathcal{H} \to j_!* = \mathcal{F} $$ i.e., we obtain a functor $j'_! : \mathop{\textit{Sh}}\nolimits(\mathcal{C}/\mathcal{F}) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C})/\mathcal{F}$. An inverse to this functor is the rule which assigns to an object $\varphi : \mathcal{G} \to \mathcal{F}$ of $\mathop{\textit{Sh}}\nolimits(\mathcal{C})/\mathcal{F}$ the sheaf $$ a(\mathcal{G}/\mathcal{F}) : (U, s) \longmapsto \{t \in \mathcal{G}(U) \mid \varphi(t) = s\} $$ We omit the verification that $a(\mathcal{G}/\mathcal{F})$ is a sheaf and that $a$ is inverse to $j'_!$. $\square$

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 6568–6580 (see updates for more information).

    \begin{lemma}
    \label{lemma-localize-topos-site}
    Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$.
    Let $\mathcal{C}/\mathcal{F}$ be the category of pairs $(U, s)$ where
    $U \in \Ob(\mathcal{C})$ and $s \in \mathcal{F}(U)$. Let a covering in
    $\mathcal{C}/\mathcal{F}$ be a family $\{(U_i, s_i) \to (U, s)\}$
    such that $\{U_i \to U\}$ is a covering of $\mathcal{C}$.
    Then $j : \mathcal{C}/\mathcal{F} \to \mathcal{C}$ is a continuous
    and cocontinuous functor of sites which induces a morphism of topoi
    $j : \Sh(\mathcal{C}/\mathcal{F}) \to \Sh(\mathcal{C})$. In fact, there
    is an equivalence $\Sh(\mathcal{C}/\mathcal{F}) =
    \Sh(\mathcal{C})/\mathcal{F}$ which turns $j$ into $j_\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    We omit the verification that $\mathcal{C}/\mathcal{F}$ is a site and
    that $j$ is continuous and cocontinuous. By
    Lemma \ref{lemma-when-shriek} there exists a morphism of topoi
    $j$ as indicated, with $j^{-1}\mathcal{G}(U, s) = \mathcal{G}(U)$,
    and there is a left adjoint $j_!$ to $j^{-1}$. A morphism
    $\varphi : * \to j^{-1}\mathcal{G}$ on $\mathcal{C}/\mathcal{F}$
    is the same thing as a rule which assigns to every pair $(U, s)$ a
    section $\varphi(s) \in \mathcal{G}(U)$ compatible with restriction maps.
    Hence this is the same thing as a morphism
    $\varphi : \mathcal{F} \to \mathcal{G}$ over $\mathcal{C}$.
    We conclude that $j_!* = \mathcal{F}$. In particular, for every
    $\mathcal{H} \in \Sh(\mathcal{C}/\mathcal{F})$ there is a canonical map
    $$
    j_!\mathcal{H} \to j_!* = \mathcal{F}
    $$
    i.e., we obtain a functor
    $j'_! : \Sh(\mathcal{C}/\mathcal{F}) \to \Sh(\mathcal{C})/\mathcal{F}$.
    An inverse to this functor is the rule which assigns to an object
    $\varphi : \mathcal{G} \to \mathcal{F}$ of $\Sh(\mathcal{C})/\mathcal{F}$ the
    sheaf
    $$
    a(\mathcal{G}/\mathcal{F}) :
    (U, s) \longmapsto \{t \in \mathcal{G}(U) \mid \varphi(t) = s\}
    $$
    We omit the verification that $a(\mathcal{G}/\mathcal{F})$ is a sheaf
    and that $a$ is inverse to $j'_!$.
    \end{proof}

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